Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19420 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,15)(13,18)(14,17)(19,20,25,28)(21,23,27,24)(22,29,32,30)(26,33,34,31);(3,5,7,15,8,13,6,14)(4,9,10,12,11,17,16,18)(19,21,23,28,25,27,24,20)(22,30,31,26,32,29,33,34);(5,13)(9,17)(12,18)(14,15)(19,22,25,32)(20,26,28,34)(21,29,27,30)(23,31,24,33);(3,6,8,7)(4,10,11,16)(5,14,13,15)(9,12,17,18)(19,23,25,24)(20,27,28,21)(22,31,32,33)(26,30,34,29)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261691);